## Optical Absorption

Another way to probe the electronic properties of SWNTs is through their optical absorption spectra. This technique is based on the well known phenomenon of light absorption

1400 1500 1600 1700

wavenumber(cm-1)

Figure 14. Resonant Raman spectra in the tangential mode (G mode) obtained using three different laser excitation energies, for SWNT samples produced by CO disproportionation at 950 °C. Reprinted with permission from [88], J. E. Herrera et al., J. Nanotech., in press. © American Scientific Publishers.

by a chromophore group present in the sample. In the case of SWNTs, when the light sent to the sample matches the energy of an allowed electronic transition between van Hove singularities, light in the visible and near infrared range can be absorbed. The observed absorption peaks are identified with interband transitions Eu{dt) between the ith van Hove singularity in the valence band (occupied states) to the ith singularity in the conduction band (empty states). Figure 15 shows typical optical absorption spectra of samples synthesized by laser ablation using different catalysts [78, 138]. For the sample obtained by laser ablation using a NiY catalyst (with an average diameter of 1.5 nm) three strong absorption bands at 0.68, 1.2, and 1.7 eV were observed which are related to interband transitions between the first and second van Hove singularities (E^ and E|2) for semiconducting nanotubes and E* for metallic nanotubes, as can be predicted by observing Figure 8. Notice that the optical absorption spectra for the SWNTs obtained using a RhPd catalyst correspond to SWNTs with much smaller diameters (diameters distributed from 0.68 to 1.00 nm) so the theoretical calculations depicted in Figure 8 predict that the optical absorption peaks should move to higher energies, as is experimentally verified [139].

As mentioned above, the same calculations that relate SWNT diameter with the energy of the optical interband transitions can be used to predict the positions of the absorption bands. The selection rules predict that the optical absorption spectra of SWNT will be dominated by transitions between peaks in the density of states of the valence and conduction bands, with momentum conservation only allowing transitions between pairs of singularities which are symmetrically placed with respect to the Fermi level. Thus, following the van Hove singularities, the energies of the optical transitions in SWNTs are also inversely proportional to

the SWNT diameter. For the first two allowed optical transitions in semiconducting SWNTs, the energy of the optical transitions has been proposed to be proportional to [140]

where y is the tight binding nearest neighbor w-w overlap integral, which can be calculated from the Raman spectra. For the metallic nanotubes, the energies of the optical transitions have been proposed to be EMM = 6a0y/d [81]. However, these calculations were shown to be inaccurate; indeed more recently it has been pointed out that the density of states of the metallic SWNT is also chirality dependent due to a trigonal warp effect [141, 142]. This leads to a splitting of the singularities in metallic nanotubes, which is maximum for the zigzag variety. Therefore, a more careful accounting of the electronic transitions has to be carried out in order to obtain a more accurate model.

One of the great advantages of optical absorption is the possibility of achieving high-energy resolution, which allows the identification of fine structure within the individual absorption features [83]. This fine structure is related to individual SWNTs, or groups of SWNTs with similar diameters. Moreover, it has been proposed that the analysis of such data would offer information as to whether the formation process of SWNT leads to the existence of preferred wrapping angles in the nanotube vector map [56, 81, 83].

In general, the results obtained by optical absorption must be compared with the information obtained by Raman spec-troscopy. Since in optical absorption there is no resonance limitation as in Raman spectroscopy, in which a narrow set of SWNTs is probed, the interpretation of the optical absorption information is more straightforward. A true diameter distribution of the sample can be obtained from the absorption spectra. Two different approaches have been proposed to obtain the parameters for the diameter distribution. The first approach is based on the superposition of the full set of DOS functions to represent the quasi-continuous distribution of nanotubes [82]. In this approach, each nano-tube is weighted with a Gaussian factor and by using the assumption of constant matrix elements the absorption is obtained from a(E)a^2 exp dn, m do

This expression renders characteristic peaks for the optical transitions between the van Hove singularities. Using a fixed value of y for the w-w tight binding overlap integral and a similar scaling for the first two transitions, d0 and a can be fitted to the experimental results. Figure 16 shows how this approach yields information on the diameter distribution for two different SWNT samples. The results obtained using Eq. (6) are shown as dashed lines while the experimental data are superimposed as a continuous line. The two SWNT samples were obtained by laser ablation using a Ni-Co catalyst and by arc discharge using a Ni-Y catalyst. The results of the fitting were d0 = 1.38 nm and a = 0.11 nm for the former and d0 = 1.47 nm and a = 0.12 nm for the latter. These results are in good agreement with the

Figure 16. Optical absorption for four different SWNT samples after subtraction of the background (full line). Calculated absorption spectra are shown as dashed lines. The upper two dashed spectra hold for a calculation using the full set of DOS functions. The lower two dashed spectra hold for the approximation according to Eq. (6). Reprinted with permission from [82], H. Kuzmany et al., Eur. Phys. J. B 22, 307 (2001). © 2001, Springer-Verlag.

Figure 16. Optical absorption for four different SWNT samples after subtraction of the background (full line). Calculated absorption spectra are shown as dashed lines. The upper two dashed spectra hold for a calculation using the full set of DOS functions. The lower two dashed spectra hold for the approximation according to Eq. (6). Reprinted with permission from [82], H. Kuzmany et al., Eur. Phys. J. B 22, 307 (2001). © 2001, Springer-Verlag.

parameters obtained from Raman spectroscopy. The agreement was not only with respect to the RBM band position but also with respect to the line width and intensity.

An alternate analysis of the optical spectra involves an approximation where only the density of states of geometrically allowed nanotubes (chiral angle smaller than 30°) is considered [81]. Again, each nanotube is weighted with a Gaussian probability:

2a 2

8 is a small value of the order of 10 meV describing the finite resolution of the spectrometer and the width of the resonant electronic states i due to lifetime effects. Eii(n, m) is a function of the interatomic distance. The n-n overlap integral and the diameters of the SWNTs and their positions (i = 1, 2, 3) are taken from the separation between the maxima of the van Hove singularities in the SWNT electronic density of states. The reader is referred to the original work for a full description. The optical absorption data can then be fitted by varying d0 and a. It has been said that this is a better approximation than the one described in Eq. 6.

Liu et. al [81] have attempted to draw the vector map of SWNT samples by resolving the fine structure of the optical absorption spectra. Although it is recognized that an exact assignment of the (n, m) indexes cannot be done solely based on optical absorption data, it was claimed that a reasonable approximation to the chiral angle 6 could be obtained. The following equation that describes the absorption profile of a bulk sample of SWNT was proposed:

As in the case of Eq. (7), the energy positions Eii(i = 1, 2, 3) are taken from the separation between the maxima of the van Hove singularities in the SWNT electronic density of states [143].

Figure 17 shows the results of one such fitting to the first three absorption peaks of a SWNT sample synthesized by arc discharge using a NiY catalyst. In this case, the authors performed the fitting including all the (n, m) pairs in the SWNT vector map and used a value of 3.0 eV for the tight binding overlap integral. The solid line in the graph represents the experimental data after background subtraction, and the dotted line is the result of the fit. The results of the fit indicated a mean diameter of 1.39 nm and a Ad of 0.09 nm, which was in good agreement with electron diffraction results obtained on the same samples.

For the spectra of the second pair of van Hove singularities, a remarkable improvement was observed in the quality of the fit when only SWNTs with chiral angles between 15° to 30° were used. An example of this improvement is shown in Figure 18. The same improvement was observed on eight different SWNT samples (with different diameter distributions). It was concluded that SWNTs are preferentially formed closer to the armchair rather than zigzag axis during the growth process, a trend previously reported by other authors using electron diffraction and scanning tunneling microscopy [40, 42, 144].

Undoubtedly, since resonance effects do not affect optical absorption, this is a technique more reliable for the quantification of the different kinds of nanotubes present on the sample than Raman spectroscopy. However, it has been

Figure 17. Simulation of the first three optical absorption peaks of sample B with d = 1.37 nm using Eq. (8). Nanotubes of all chiral-ities are included. The solid line is the background subtracted measured spectrum; the dotted line represents the results of the simulation. Reprinted with permission from [81], X. Liu et al., Phys. Rev. B 66, 045411 (2002). © 2002, American Physical Society.

Figure 17. Simulation of the first three optical absorption peaks of sample B with d = 1.37 nm using Eq. (8). Nanotubes of all chiral-ities are included. The solid line is the background subtracted measured spectrum; the dotted line represents the results of the simulation. Reprinted with permission from [81], X. Liu et al., Phys. Rev. B 66, 045411 (2002). © 2002, American Physical Society.

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